Simultaneous savings in bandwidth and energy using waveform design in presence of interference and noise

ABSTRACT

A transmit signal is output from a transmitter towards a target and towards interference. A combination signal is received at a receiver, wherein the combination signal includes the transmit signal modified by interacting with the target and the interference along with noise. The receiver has a filter having a transfer function and the filter acts on the combination signal to form a receiver output signal having a receiver output signal waveform. The receiver output signal has a receiver output signal waveform that describes an output signal to interference to noise ratio (SINR) performance. Bandwidth and signal energy of the transmit signal are reduced simultaneously by modifying the transmit signal waveform and receiver output signal waveforms without sacrificing the output SINR performance level.

CROSS REFERENCE TO RELATED APPLICATION(S)

The present application is a continuation in part of and claims the priority of U.S. patent application Ser. No. 11/623,965, titled “APPARATUS AND METHOD FOR PROVIDING ENERGY-BANDWIDTH TRADEOFF AND WAVEFORM DESIGN IN INTERFERENCE AND NOISE”, filed on Jan. 17, 2007, inventor UNNIKRISHNA SREEDHARAN PILLAI.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

The present invention is based upon work supported and/or sponsored by the Air Force Research Laboratory (AFRL), Rome, N.Y., under contract No. FA8750-06-C-0202.

FIELD OF INVENTION

The invention relates to techniques concerning transmitter-receiver waveform design methods that are applicable to radar, sonar and wireless communications.

BACKGROUND OF INVENTION

In the general problem, a desired target is buried in both interference and noise. A transmit signal excites both the desired target and the interference simultaneously. The interference and/or interferences can be foliage returns in the form of clutter for radar, scattered returns of the transmit signal from a sea-bottom and different ocean-layers in the case of sonar, or multipath returns in a communication scene. In all of these cases, like the target return, the interference returns are also transmit signal dependent, and hence it puts conflicting demands on the receiver. In general, the receiver input is comprised of target returns, interferences and the ever present noise. The goal of the receiver is to enhance the target returns and simultaneously suppress both the interference and noise signals. In a detection environment, a decision regarding the presence or absence of a target is made at some specified instant t=t_(o) using output data from a receiver, and hence to maximize detection, the Signal power to average Interference plus Noise Ratio (SINR) at the receiver output can be used as an optimization goal. This scheme is illustrated in FIG. 1.

The transmitter output bandwidth can be controlled using a known transmitter output filter having a transfer function P₁(ω) (see FIG. 2A). A similar filter with transform characteristics P₂(ω) can be used at a receiver input 22 a shown in FIG. 1, to control the processing bandwidth as well.

The transmit waveform set f(t) at an output 10 a of FIG. 1, can have spatial and temporal components to it each designated for a specific goal. The simplest situation is that shown in FIG. 2A where a finite duration waveform f(t) of energy E is to be designed. Thus

$\begin{matrix} {{\int_{0}^{T_{o}}{{{f(t)}}^{2}\ {\mathbb{d}t}}} = {E.}} & (1) \end{matrix}$

Usually, transmitter output filter 12 characteristics P₁(ω), such as shown in FIG. 2B, are known and for design purposes, it is best to incorporate the transmitter output filter 12 and the receiver input filter (which may be part of receiver 22) along with the target and clutter spectral characteristics.

Let q(t)

Q(ω) represent the target impulse response and its transform. In general q(t) can be any arbitrary waveform. Thus the modified target that accounts for the target output filter has transform P₁(ω)Q(ω) etc. In a linear domain setup, the transmit signal f(t) interacts with the target q(t), or target 14 shown in FIG. 1, to generate the output below (referred to in S. U. Pillai, H. S. Oh, D. C. Youla, and J. R. Guerci, “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, IEEE Transactions on Information Theory, Vol. 46, No. 2, pp. 577-584, March 2000 and J. R. Guerci and S. U. Pillai, “Theory and Application of Optimum Transmit-Receive Radar,” IEEE International Radar Conference, Alexandria Va., May 2000, pp. 705-710.):

$\begin{matrix} {{s(t)} = {{{f(t)}*{q(t)}} = {\int_{0}^{T_{o}}{{f(\tau)}{q\left( {t - \tau} \right)}\ {\mathbb{d}\tau}}}}} & (2) \end{matrix}$ that represents the desired signal.

The interference returns are usually due to the random scattered returns of the transmit signal from the environment, and hence can be modeled as a stochastic signal w_(C)(t) that is excited by the transmit signal f(t). If the environment returns are stationary, then the interference can be represented by its power spectrum G_(c)(ω). This gives the average interference power to be G_(c)(ω)|F(ω)|². Finally let n(t) represent the receiver 22 input noise with power spectral density G_(n)(ω). Thus the receiver input signal at input 22 a equals r(t)=s(t)+w _(c)(t)*f(t)+n(t),  (3) and the input interference plus noise power spectrum equals G _(I)(ω)=G _(c)(ω)|F(ω)|² +G _(n)(ω).  (4) The received signal is presented to the receiver 22 at input 22 a with impulse response h(t). The simplest receiver is of the noncausal type.

With no restrictions on the receiver 22 of FIG. 1, its output signal at output 22 b in FIG. 1, and interference noise components are given by

$\begin{matrix} {{{y_{S}(t)} = {{{s(t)}*{h(t)}} = {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{S(\omega)}{H(\omega)}{\mathbb{e}}^{{j\omega}\; t}\ {\mathbb{d}\omega}}}}}}{and}} & (5) \\ {{y_{n}(t)} = {\left\{ {{{w_{c}(t)}*{f(t)}} + {n(t)}} \right\}*{{h(t)}.}}} & (6) \end{matrix}$ The output y_(n)(t) represents a second order stationary stochastic process with power spectrum below (referred to in the previous publications and in Athanasios Papoulis, S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill Higher Education, New York 2002): G _(o)(ω)=(G _(c)(ω)|F(ω)|² +G _(n)(ω))|H(ω)|²  (7) and hence the total output interference plus noise power is given by

$\quad\begin{matrix} \begin{matrix} {\sigma_{I + N}^{2} = {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{G_{O}(\omega)}\ {\mathbb{d}\omega}}}}} \\ {= {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{\left( {{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}} \right){{H(\omega)}}^{2}\ {{\mathbb{d}\omega}.}}}}} \end{matrix} & (8) \end{matrix}$ Referring back to FIG. 1, the signal component y_(s)(t) in equation (5) at the receiver output 22 b needs to be maximized at the decision instant t_(o) in presence of the above interference and noise. Hence the instantaneous output signal power at t=t_(o) is given by the formula below shown in S. U. Pillai, H. S. Oh, D. C. Youla, and J. R. Guerci, “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, IEEE Transactions on Information Theory, Vol. 46, No. 2, pp. 577-584, March 2000, which is incorporated by reference herein:

$\begin{matrix} {P_{O} = {{{y_{S}\left( t_{O} \right)}}^{2} = {{{\frac{1}{\ {2\pi}\;}{\int_{- \infty}^{+ \infty}{{S(\omega)}{H(\omega)}{\mathbb{e}}^{{j\omega}\; t_{o}}\ {\mathbb{d}\omega}}}}}^{2}.}}} & (9) \end{matrix}$ This gives the receiver output SINR at t=t_(o) be the following as specified in Guerci et. al., “Theory and Application of Optimum Transmit-Receive Radar”, pp. 705-710; and Pillai et. al., “Optimum Transmit-Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise”, incorporated herein by reference:

$\begin{matrix} {{SINR} = {\frac{P_{O}}{\sigma_{I + N}^{2}} = {\frac{{{\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{S(\omega)}{H(\omega)}{\mathbb{e}}^{{j\omega}\; t_{o}}\ {\mathbb{d}\omega}}}}}^{2}}{\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{G_{I}(\omega)}{{H(\omega)}}^{2}\ {\mathbb{d}\omega}}}}.}}} & (10) \end{matrix}$ We can apply Cauchy-Schwarz inequality in equation (10) to eliminate H(ω). This gives

$\begin{matrix} {{{SINR} \leq {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{\frac{{{S(\omega)}}^{2}}{G_{I}(\omega)}\ {\mathbb{d}\omega}}}}} = {{\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{\frac{{{Q(\omega)}}^{2}{{F(\omega)}}^{2}}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}\ {\mathbb{d}\omega}}}} = {{SINR}_{\max}.}}} & (11) \end{matrix}$ Thus the maximum obtainable SINR is given by equation (11), and this is achieved if and only if the following equation referred to in previous prior art publications, is true:

$\quad\begin{matrix} \begin{matrix} {{H_{opt}(\omega)} = {\frac{S^{*}(\omega)}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}{\mathbb{e}}^{{- {j\omega}}\; t_{o}}}} \\ {= {\frac{{Q^{*}(\omega)}{F^{*}(\omega)}}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}{{\mathbb{e}}^{{- {j\omega}}\; t_{o}}.}}} \end{matrix} & (12) \end{matrix}$ In (12), the phase shift e^(−1ωt) _(o) can be retained to approximate causality for the receiver waveform. Interestingly even with a point target (Qω≡1), flat noise (G_(n)(ω)=σ_(n) ²), and flat clutter (G_(c)(ω)=σ_(c) ²), the optimum receiver is not conjugate-matched to the transmit signal, since in that case from equation (12) we have the following formula given by Pillai et. al., “Optimum Transmit-Receiver Design in the Presence of Signal -Dependent Interference and Channel Noise”, incorporated herein by reference, Papoulis, “Probability, Random Variables and Stochastic Processes”, and H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I, New York: John Wiley and Sons, 1968, incorporated by reference:

$\begin{matrix} {{H_{opt}(\omega)} = {{\frac{F^{*}(\omega)}{{\sigma_{c}^{2}{{F(\omega)}}^{2}} + \sigma_{n}^{2}}{\mathbb{e}}^{{- {j\omega}}\; t_{o}}} \neq {{F^{*}(\omega)}{{\mathbb{e}}^{{- {j\omega}}\; t_{o}}.}}}} & (13) \end{matrix}$ Prior Art Transmitter Waveform Design

When the receiver design satisfies equation (12), the output SINR is given by the right side of the equation (11), where the free parameter |F(ω)|² can be chosen to further maximize the output SINR, subject to the transmit energy constraint in (1). Thus the transmit signal design reduces to the following optimization problem:

Maximize

$\begin{matrix} {{{SINR}_{\max} = {\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{\frac{{{Q(\omega)}}^{2}{{F(\omega)}}^{2}}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}\ {\mathbb{d}\omega}}}}},} & (14) \end{matrix}$ subject to the energy constraint

$\begin{matrix} {{\int_{0}^{T_{o}}{{{f(t)}}^{2}\ {\mathbb{d}t}}} = {{\frac{1}{2\pi}{\int_{- \infty}^{+ \infty}{{{F(\omega)}}^{2}\ {\mathbb{d}\omega}}}} = {E.}}} & (15) \end{matrix}$ To solve this new constrained optimization problem, combine (14)-(15) to define the modified Lagrange optimization function (referred to in T. Kooij, “Optimum Signal in Noise and Reverberation”, Proceeding of the NATO Advanced Study Institute on Signal Processing with Emphasis on Underwater Acoustics, Vol. I, Enschede, The Netherlands, 1968.)

$\begin{matrix} {\Lambda = {\int_{- \infty}^{+ \infty}{\left\{ {\frac{{{Q(\omega)}}^{2}{y^{2}(\omega)}}{{{G_{c}(\omega)}{y^{2}(\omega)}} + {G_{n}(\omega)}} - {\frac{1}{\lambda^{2}}{y^{2}(\omega)}}} \right\}\ {\mathbb{d}\omega}}}} & (16) \end{matrix}$ where y(ω)=|F(ω)|  (17) is the free design parameter. From (16) (17),

$\frac{\partial\Lambda}{\partial y} = 0$ gives

$\begin{matrix} {\frac{\partial{\Lambda(\omega)}}{\partial y} = {{2{y(\omega)}\left\{ {\frac{{G_{n}(\omega)}{{Q(\omega)}}^{2}}{\left\{ {{{G_{c}(\omega)}{y^{2}(\omega)}} + {G_{n}(\omega)}} \right\}^{2}} - \frac{1}{\lambda^{2}}} \right\}} = 0.}} & (18) \end{matrix}$ where Λ(ω) represents the quantity within the integral in (16). From (18), either

$\begin{matrix} {{{y(\omega)} = 0}{or}} & (19) \\ {{{\frac{{G_{n}(\omega)}{{Q(\omega)}}^{2}}{\left\{ {{{G_{c}(\omega)}{y^{2}(\omega)}} + {G_{n}(\omega)}} \right\}^{2}} - \frac{1}{\lambda^{2}}} = 0},{{which}\mspace{14mu}{gives}}} & (20) \\ {{y^{2}(\omega)} = \frac{\sqrt{G_{n}(\omega)}\left( {{\lambda{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)}} & (21) \end{matrix}$ provided y²(ω)>0. See T. Kooij cited above incorporated by reference herein.

SUMMARY OF THE INVENTION

One or more embodiments of the present invention provide a method and an apparatus for transmitter-receiver design that enhances the desired signal output from the receiver while minimizing the total interference and noise output at the desired decision making instant. Further the method and apparatus of an embodiment of the present invention can be used for transmit signal energy-bandwidth tradeoff. As a result, transmit signal energy can be used to tradeoff for “premium” signal bandwidth without sacrificing performance level in terms of the output Signal to Interference plus Noise power Ratio (SINR). The two designs—before and after the tradeoff—will result in two different transmitter-receiver pairs that have the same performance level. Thus a design that uses a certain energy and bandwidth can be traded off with a new design that uses more energy and lesser bandwidth compared to the old design. In many applications such as in telecommunications, since the available bandwidth is at premium, such a tradeoff will result in releasing otherwise unavailable bandwidth at the expense of additional signal energy. The bandwidth so released can be used for other applications or to add additional telecommunications capacity.

In addition, a new method for transmitter-receiver design that can be used for simultaneous savings of transmit signal energy and bandwidth compared to any prior art waveform without sacrificing the performance level is described here. Thus, simultaneous savings in transmit signal energy as well as the more “premium” signal bandwidth, compared to prior art transmit waveforms, can be realized without sacrificing the system performance level in terms of the output Signal to Interference plus Noise power Ratio (SINR). The new design procedure in fact leads to various new transmitter -receiver waveform designs that trade off bandwidth and energy for the same performance level.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a diagram of a system, apparatus, and/or method including a transmitter, a transmitter output filter, a receiver, a target, interference, noise, and a switch;

FIG. 2A shows a prior art graph of a prior art transmitter signal versus time, wherein the transmitter signal is output from a transmitter, such as in FIG. 1;

FIG. 2B shows a prior art graph of a possible frequency spectrum of a known transmitter output filter, such as in FIG. 1;

FIG. 3A shows a graph of target transfer function magnitude response versus frequency;

FIG. 3B shows a graph of target transfer function magnitude response versus frequency;

FIG. 3C shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency;

FIG. 3D shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency;

FIG. 4A shows graphs of three different target transfer function magnitude responses versus frequency;

FIG. 4B shows a graph of noise power spectrum versus frequency;

FIG. 4C shows a graph of clutter power spectrum versus frequency;

FIG. 4D shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency;

FIG. 4E shows a graph of transmitter threshold energy versus bandwidth;

FIG. 4F shows a graph of signal to inference plus noise ratio (SINR) versus bandwidth;

FIG. 5A shows graphs of three different target transfer function magnitude responses versus frequency;

FIG. 5B shows a graph of noise power spectrum versus frequency;

FIG. 5C shows a graph of clutter power spectrum versus frequency;

FIG. 5D shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency;

FIG. 5E shows a graph of transmitter threshold energy versus bandwidth;

FIG. 5F shows a graph of signal to inference plus noise ratio (SINR) versus bandwidth;

FIG. 6A shows a graph of signal to interference plus noise ratio versus energy for a resonant target shown in FIG. 5A (solid line);

FIG. 6B shows a graph of signal to interference plus noise ratio versus energy for a low pass target shown in FIG. 5A (dashed line);

FIG. 6C shows a graph of signal to interference plus noise ratio versus energy for a flat target shown in FIG. 5A (dotted line);

FIG. 7 shows a graph of signal to interference plus noise ratio versus energy and the Bandwidth-Energy swapping design;

FIG. 8A shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point A in FIG. 7;

FIG. 8B shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point B in FIG. 7;

FIG. 8C shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point C in FIG. 7;

FIG. 9 is a graph of realizable bandwidth savings versus operating bandwidth;

FIG. 10 shows simultaneous energy and bandwidth saving design—graphs of SINR versus energy for optimum waveform and chirp waveform;

FIG. 11 shows a prior art graph of a possible frequency spectrum of a bandlimiting filter;

FIG. 12A shows a graph of a target transfer function magnitude responses versus frequency;

FIG. 12B shows a graph of noise power spectrum versus frequency;

FIG. 12C shows a graph of clutter power spectrum versus frequency;

FIG. 12D shows a graph of the square root of the noise power spectrum divided by target transfer function magnitude response that has been modified by the transmitter output filter versus frequency;

FIG. 13A shows graphs of simultaneous energy and bandwidth saving design—SINR versus energy for optimum waveform and chirp using target, noise and clutter spectra shown in FIG. 12 and transmitter output filter given by FIG. 11;

FIG. 13B shows graphs of simultaneous energy and bandwidth saving design—SINR versus Energy for optimum waveform and chirp using target-enlarged version of FIG. 13A;

FIG. 14A shows a graph of the transform of the chirp transmit signal magnitude versus frequency corresponding to the design point A in FIG. 13;

FIG. 14B shows a graph of the transform of the optimum transmitter signal magnitude versus frequency corresponding to the design point C in FIG. 13;

FIG. 14C shows a graph of the transform of the optimum transmitter signal magnitude versus frequency corresponding to the design point F in FIG. 13;

FIG. 14D shows a graph of the transform of the optimum transmitter signal magnitude versus frequency corresponding to the design point A in FIG. 13;

FIG. 15A shows a graph of the chirp transmit signal versus time associated with FIG. 14A and the design point A in FIG. 13;

FIG. 15B shows a graph of the optimum transmit signal versus time associated with FIG. 14B and the design point C in FIG. 13;

FIG. 15C shows a graph of the optimum transmit signal versus time associated with FIG. 14C and the design point F in FIG. 13;

FIG. 15D shows a graph of the optimum transmit signal versus time associated with FIG. 14D and the design point A in FIG. 13; and

FIG. 16 shows a diagram of a system, apparatus, and/or method in accordance with another embodiment of the present invention.

DETAILED DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a diagram 1 of a system, apparatus, and/or method, including a transmitter 10, a transmitter output filter 12, a target 14, interference 16, noise 18, a summation block 20, receiver 22, and a switch 24. The present invention, in one or more embodiments, provides a new method and apparatus, by selecting a particular transmit signal f(t), to be output from transmitter 10, and a type of receiver or receiver transfer function for receiver 22 in accordance with criteria to be discussed below.

The transmitter 10 transmits an output signal f(t) at its output 10 a and supplies this signal to the transmitter output filter 12. As remarked earlier, for design purposes, the transmitter output filter 12 can be lumped together with the target transfer function ?? as well as the interference spectrum. The transmit signal f(t) passes through the airwaves and interacts with a target 14 and interference 16. The target-modified as well as the clutter -modified (or interference modified) versions of the transmit signal f(t) are supplied to the summation block 18 along with receiver noise 18. The summation block 18 may simply be used for description purposes to indicate that the target modified, clutter modified, and noise signals combine together. A combination signal is supplied to receiver 22 at its input 22 a. The receiver 22 applies a transfer function H(ω) (which will be determined and/or selected by criteria of an embodiment of the present invention, to be described below) and a modified combination signal is provided at a receiver output 22 b. The output is accessed at time t=t₀ by use of switch 24.

FIG. 2A shows a prior art graph of a prior art transmitter output signal f(t) versus time. The signal used here is arbitrary.

FIG. 2B shows a prior art graph of a frequency spectrum of the transmitter output filter 12 of FIG. 1.

FIG. 3A shows a typical graph of a target transfer function magnitude response for target 14 versus frequency; target as appearing in (14)-(21).

FIG. 3B shows a typical graph of target transfer function magnitude response for target 14 versus frequency; target as appearing in (14)-(21).

FIG. 3C shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency; as in right side of equation (23).

FIG. 3D shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency; as in right side of equation (23).

FIG. 4A shows graphs of three different target transfer function magnitude responses versus frequency; target as appearing in (14)-(21).

FIG. 4B shows a graph of noise power spectrum versus frequency as appearing in equations (14)-(23).

FIG. 4C shows a graph of clutter power spectrum versus frequency as appearing in equations (14)-(23).

FIG. 4D shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency as in right side of equation (23).

FIG. 4E shows a graph of transmitter threshold energy versus bandwidth using equation (26).

FIG. 4F shows a graph of signal to inference plus noise ratio versus bandwidth using equations (27)-(31).

FIG. 5A shows graphs of three different target transfer function magnitude responses versus frequency; target as appearing in (14)-(21).

FIG. 5B shows a graph of noise power spectrum versus frequency as appearing in equations (14)-(23).

FIG. 5C shows a graph of clutter power spectrum versus frequency as appearing in equations (14)-(23).

FIG. 5D shows a graph of noise power spectrum divided by target transfer function magnitude response versus frequency as in right side of equation (23).

FIG. 5E shows a graph of transmitter threshold energy versus bandwidth using equation (26).

FIG. 5F shows a graph of signal to inference plus noise ratio versus bandwidth using equations (27)-(31).

FIG. 6A shows a graph of signal to interference plus noise ratio versus energy for a resonant target shown in FIG. 5A (solid line) using equations (34)-(35).

FIG. 6B shows a graph of signal to interference plus noise ratio versus energy for a low pass target shown in FIG. 5A (dashed line) using equations (34)-(35).

FIG. 6C shows a graph of signal to interference plus noise ratio versus energy for a flat target shown in FIG. 5A (dotted line) using equations (34)-(35).

FIG. 7 shows a graph of signal to interference plus noise ratio versus energy; generated using equations (39), (48), and (51).

FIG. 8A shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point A in FIG. 7 generated using (42).

FIG. 8B shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point B in FIG. 7 generated using (42).

FIG. 8C shows a graph of the transform of the transmitter signal versus frequency corresponding to the design point C in FIG. 7 generated using (42) for a third energy condition.

FIG. 9 is a realizable bandwidth savings versus operating bandwidth generated using equation (60).

FIG. 10 shows a typical graph of simultaneous energy and bandwidth saving design—graphs of SINR versus energy for optimum waveform and chirp waveform;

FIG. 11 shows a prior art graph of a possible frequency spectrum of a bandlimiting filter used in FIG. 12D;

FIG. 12A shows a graph of target transfer function magnitude responses versus frequency; target as appearing in equations (29)-(31) and (67)-(71).

FIG. 12B shows a graph of noise power spectrum versus frequency as appearing in equations (29)-(31) and (67)-(71).

FIG. 12C shows a graph of clutter power spectrum versus frequency as appearing in equations (29)-(31) and (67)-(71).

FIG. 12D shows a graph of the square root of the noise power spectrum divided by target transfer function magnitude response that has been modified by the transmitter output filter versus frequency as appearing in right side of equation (23).

FIG. 13A shows graphs of simultaneous energy and bandwidth saving design—SINR versus Energy for optimum waveform and chirp using target, noise and clutter spectra shown in FIG. 12 and transmitter output filter given by FIG. 11; generated using equations (39), (48), and (51) for the optimum waveforms 1001, 1002 and 1004 and (63) for the chirp waveform 1002.

FIG. 13B shows graphs of simultaneous energy and bandwidth saving design-SINR versus Energy for optimum waveform and chirp using target-Enlarged version of FIG. 13A; generated using equations (39), (48), and (51) for the optimum waveforms and (63) for the chirp waveform.

FIG. 14A shows a graph of the transform of the chirp transmit signal magnitude versus frequency corresponding to the design point A in FIG. 13;

FIG. 14B shows a graph of the transform of the optimum transmitter signal magnitude versus frequency corresponding to the design point C in FIG. 13 generated using equation (36);

FIG. 14C shows a graph of the transform of the optimum transmitter signal magnitude versus frequency corresponding to the design point F in FIG. 13 generated using equation (42);

FIG. 14D shows a graph of the transform of the optimum transmitter signal magnitude versus frequency corresponding to the design point A in FIG. 13 generated using equation (42);

FIG. 15A shows a graph of a typical chirp transmit signal versus time associated with FIG. 14A and the design point A in FIG. 13;

FIG. 15B shows a graph of the optimum transmit signal versus time associated with FIG. 14B and the design point C in FIG. 13; generated using equations (76)-(79);

FIG. 15C shows a graph of the optimum transmit signal versus time associated with FIG. 14C and the design point F in FIG. 13; generated using equations (76)-(79);

FIG. 15D shows a graph of the optimum transmit signal versus time associated with FIG. 14D and the design point A in FIG. 13; generated using equations (76)-(79); and

FIG. 16 shows a diagram 1600 of a system, apparatus, and/or method in accordance with another embodiment of the present invention. The diagram 1600 shows a data input device 1601, a computer processor 1610, a transmitter output filter 12, a target 14, interference 16, noise 18, a summation block 20, receiver 22, and a switch 24. The computer processor 1610 may implement one or more methods in accordance with one or more embodiments of the present invention. The computer processor 1610 may use the equations in (67)-(80) to determine an appropriate transmit signal to be provided at the output 10 a. Appropriate input parameters for using in calculations, such as calculations using the equations in (67)-(80) may be provided by data input device 1601. The data input device 1601 may be an interactive device such as a computer keyboard or computer mouse which an operator can use to input data or input parameters. The data input device 1601 may also be a computer data base which stores various data or input parameters. The computer processor 1610 may receive parameters from the data input device 1601 at its input 1610 a and may generate a desired transmit waveform at its output. The data input device 1601 may supply required input parameters such as target transform, clutter and noise spectra, and bandwidth to the computer processor 1610. The computer processor 1610 can be a software toolbox or actual hardware implementation of the methods described previously.

Define Ω₊ as shown in FIGS. 3C and 3D to represent the frequencies over which y²(ω) in equation (21) is strictly positive, and let Ω_(o) represent the complement of Ω₊. As shown in FIGS. 3C and 3D, observe that the set Ω₊ is a function of the noise and target spectral characteristics as well as the constraint constant λ. In terms of Ω₊, we have

$\begin{matrix} {{{F(\omega)}}^{2} = \left\{ \begin{matrix} {{y^{2}(\omega)},} & {{\omega \in \Omega_{+}},} \\ {0,} & {\omega \in {\Omega_{O}.}} \end{matrix} \right.} & (22) \end{matrix}$ From (21), y²(ω)>0 over Ω₊ gives the necessary condition

$\begin{matrix} {\lambda \geq {\max\limits_{\omega \in \Omega_{+}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}} & (23) \end{matrix}$ and the energy constraint in (15) when applied to (21) gives

$\begin{matrix} {E = {{\frac{1}{2\pi}{\int_{\Omega_{+}}{{y^{2}(\omega)}\ {\mathbb{d}\omega}}}} = {{\frac{\lambda}{2\pi}{\int_{\Omega_{+}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}} - {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}}}} & (24) \end{matrix}$ or, for a given value of E, we have

$\begin{matrix} {\lambda = {\frac{E + {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}}{\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}{{{\bullet\lambda}(E)}.}}} & (25) \end{matrix}$ Clearly, λ(E) in (25) must satisfy the inequality in (23) as well. This gives rise to the concept of transmitter energy threshold that is characteristic to this design approach. Transmitter Threshold Energy

From (23)-(25), the transmit energy E must be such that λ(E) obtained from (25) should satisfy (23). If not, E must be increased to accommodate it, and hence it follows that there exists a minimum threshold value for the transmit energy below which it will not be possible to maintain |F(ω)|²>0. This threshold value is given by

$\begin{matrix} {E_{\min} = {{\left( {\max\limits_{\omega \in \Omega_{+}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}} - {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}}} & (26) \end{matrix}$ and for any operating condition, the transmit energy E must exceed E_(min). Clearly, the minimum threshold energy depends on the target, clutter and noise characteristics as well as the bandwidth under consideration. With E>E_(min), substituting (20)-(21) into the SINR_(max) in (14) we get

$\quad\begin{matrix} \begin{matrix} {\quad{{SINR}_{\max} = {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{{{Q(\omega)}}^{2}{y^{2}(\omega)}}{\lambda\sqrt{G_{n}(\omega)}{{Q(\omega)}}}\ {\mathbb{d}\omega}}}}}} \\ {= {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{{Q(\omega)}}{{\lambda(E)}\sqrt{G_{n}(\omega)}}\frac{\sqrt{G_{n}(\omega)}\left( {{{\lambda(E)}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}} \\ {\frac{1}{2\pi}{\int_{\Omega_{+}}{\left( {{{Q(\omega)}} - \frac{\sqrt{G_{n}(\omega)}}{\lambda(E)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}\ {{\mathbb{d}\omega}.}}}} \end{matrix} & (27) \end{matrix}$ Finally making use of (25), the output SINR_(max) becomes

$\begin{matrix} {\begin{matrix} {{SINR}_{1} = {{\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}} - \frac{\left( {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{{{Q(\omega)}}\sqrt{G_{n}(\omega)}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}} \right)^{2}}{E + {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}}}} \\ {= {{a - \frac{c}{\lambda(E)}} = {a - \frac{c^{2}}{E + b}}}} \\ {= \frac{{aE} + \left( {{ab} - c^{2}} \right)}{E + b}} \end{matrix}{where}} & (28) \\ {{a = {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},} & (29) \\ {{b = {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},{and}} & (30) \\ {c = {\frac{1}{2\pi}{\int_{\Omega_{+}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {{\mathbb{d}\omega}.}}}}} & (31) \end{matrix}$ Notice that ab−c²≧0. (This was published in Waveform Diversity and Design conference, Kauai, Hi., January 2006).

The optimization problem in (14)-(15) can be restated in term of Ω₊ as follows: Given Q(ω), G_(c)(ω), G_(n)(ω) and the transmit energy E, how to partition the frequency axis into an “operating band” Ω₊ and a “no show” band Ω_(o) so that λ₊ obtained from (25) satisfies (23) and SINR_(max) in (27)-(28) is also maximized. In general maximization of SINR_(max) in (27)-(28) over Ω₊ is a highly nonlinear optimization problem for arbitrary Q(ω), G_(c)(ω) and G_(n)(ω).

In what follows a new approach to this problem is presented.

AN EMBODIMENT OF THE PRESENT INVENTION—DESIRED BAND APPROACH

One approach in this situation is to make use of the “desired frequency band” of interest B_(o) this is usually suggested by the target response Q(ω) (and the transmitter output filter) to determine the operating band Ω₊. The desired band B_(o) can represent a fraction of the total available bandwidth, or the whole bandwidth itself. The procedure for determining Ω₊ is illustrated in FIGS. 3A-3C and FIGS. 3B-3D for two different situations. In FIGS. 3A-3D, the frequency band B_(o) represents the desired band, and because of the nature of the noise and clutter spectra, it may be necessary to operate on a larger region Ω₊ in the frequency domain. Thus the desired band B_(o) is contained always within the operating band Ω₊. To determine Ω₊, using equation (23) we project the band B_(o) onto the spectrum √{square root over (G_(n)(ω))}/|Q(ω)| and draw a horizontal line corresponding to

$\begin{matrix} {\lambda_{B_{o}} = {\max\limits_{\omega \in B_{o}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}} & (32) \end{matrix}$ as shown there. Define Ω₊(B_(o)) to represent the frequency region where

$\begin{matrix} {{\omega \in {{{\Omega_{+}\left( B_{O} \right)}\text{:}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \leq \lambda_{B_{o}}}} = {\max\limits_{\omega \in B_{o}}{\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}.}}} & (33) \end{matrix}$ This procedure can give rise to two situations as shown in FIG. 3A and FIG. 3B. In FIG. 3A, the operating band Ω₊(B_(o)) coincides with the desired band B_(o) as shown in FIG. 3C, whereas in FIG. 3B, the desired band B_(o) is a subset of Ω₊(B_(o)) as seen from FIG. 3D.

Knowing Ω₊(B_(o)), one can compute λ=λ(E) with the help of equation (25) over that region, and examine whether λ so obtained satisfies (23). If not, the transmitter energy E is insufficient to maintain the operating band Ω₊(B_(o)) given in (33), and either E must be increased, or Ω₊(B_(o)) must be decreased (by decreasing B_(o)) so that (23) is satisfied. Thus for a given desired band B_(o) (or an operating band Ω₊(B_(o))), as remarked earlier, there exists a minimum transmitter threshold energy E_(B) _(o) , below which it is impossible to maintain |F(ω)|²>0 over that entire operating band.

Threshold Energy

From equations (24) and (32), we obtain the minimum transmitter threshold energy in this case to be the following (as shown in S. U. Pillai, KeYong Li and B. Himed, “Energy Threshold Constraints in Transmit Waveform Design,” 2006 international Waveform Diversity & Design Conference, Kauai, Hi., Jan. 22-27, 2006.)

$\begin{matrix} {E_{B_{o}} = {{{\frac{\lambda_{B_{o}}}{2\pi}{\int_{\Omega_{+}{(B_{o})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}} - {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{o})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}} = {{{\lambda_{B_{o}}c_{o}} - b_{o}} > 0.}}} & (34) \end{matrix}$ With E≧E_(B) _(o) , the SINR_(max) in (28) can be readily computed. In particular with E=E_(B) _(o) , we get

$\begin{matrix} {{SINR}_{1} = {{{SINR}_{1}\left( B_{o} \right)} = {a_{o} - {\frac{c_{o}^{2}}{E_{B_{o}} + b_{o}}.}}}} & (35) \end{matrix}$ Here a_(o), b_(o) and c_(o) are as given in (29)-(31) with Ω₊ replaced by Ω₊(B_(o)). Eq. (35) represents the performance level for bandwidth B_(o) using its minimum threshold energy. From (21), we also obtain the optimum transmit signal transform corresponding to energy E_(B) _(o) to be

$\quad\begin{matrix} \begin{matrix} {{{F(\omega)}}^{2} = \left\{ \begin{matrix} {\frac{\sqrt{G_{n}(\omega)}\left( {{\lambda_{B_{o}}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right.}{G_{c}(\omega)},} & {\omega \in {\Omega_{+}\left( B_{o} \right)}} \\ {0,} & {\omega \in \Omega_{o}} \end{matrix} \right.} \\ {= \left\{ {\begin{matrix} {{\sqrt{G_{n}(\omega)}\left( {\left( {\max\limits_{\omega \in B_{o}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right) - \frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}},} \\ {\omega \in {\Omega_{+}\left( B_{o} \right)}} \\ {0,} \\ {\omega \in \Omega_{o}} \end{matrix}.} \right.} \end{matrix} & (36) \end{matrix}$ To summarize, to maintain a given desired band B_(o), there exists an operating band Ω₊(B_(o))≧B_(o) over which |F(ω)|²>0 and to guarantee this, the transmit energy must exceed a minimum threshold value E_(B) _(o) given by (34).

FIGS. 4A-F shows the transmitter threshold energy E in (34) and the corresponding SINR in (35) as a function of the desired bandwidth B_(o) for various target, clutter, and noise spectra. Target to noise ratio (TNR) is set at 0 dB, and the clutter to noise power ratio (CNR) is set at 20 dB here. The total noise power is normalized to unity. The desired bandwidth B_(o) is normalized with respect to the maximum available bandwidth (e.g., carrier frequency).

In FIGS. 4A-F, the noise and clutter have flat spectra and for the highly resonant target (solid line), the required minimum energy threshold and the SINR generated using (34)-(35) reach a saturation value for small values of the bandwidth. In the case of the other two targets, additional bandwidth is required to reach the maximum attainable SINR. This is not surprising since for the resonant target, a significant portion of its energy is concentrated around the resonant frequency. Hence once the transmit signal bandwidth reaches the resonant frequency, it latches onto the target features resulting in maximum SINR at a lower bandwidth.

FIGS. 5A-F show results for a new set of clutter and noise spectra as shown there; the transmitter threshold energy E in (34) and the corresponding SINR in (35) as a function of the desired bandwidth B_(o) show similar performance details.

From FIG. 5F, in the case of the resonant target (solid curve) the SINR reaches its peak value resulting in saturation even when B_(o) is a small fraction of the available bandwidth. This is because in that case, the transmit waveform is able to latch onto the dominant resonant frequency of the target. On the other extreme, when the target has flat characteristics (dotted curve), there are no distinguishing frequencies to latch on, and the transmitter is unable to attain the above maximum SINR even when B_(o) coincides with the total available bandwidth. For a low pass target (dashed curve), the transmitter is indeed able to deliver the maximum SINR by making use of all the available bandwidth.

As FIG. 3B shows, Ω₊(B_(o)) can consist of multiple disjoint frequency bands whose complement Ω_(o) represents the “no show” region. Notice that the “no show” region Ω_(o) in the frequency domain in (36) for the optimum transmit signal can be controlled by the transmit energy E in (25). By increasing E, these “no show” regions can be made narrower and this defines a minimum transmitter threshold energy E_(∞) that allows Ω₊(B_(o)) to be the entire available frequency axis. To determine E_(∞), let λ_(∞) represent the maximum in (23) over the entire frequency axis. Thus

$\begin{matrix} {{\lambda_{\infty} = {\max\limits_{{\omega } < \infty}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}},} & (37) \end{matrix}$ and let a_(∞), b_(∞), c_(∞) refer to the constants a, b, c in (29)-(31) calculated with Ω₊ representing the entire frequency axis. Then from (24)

$\begin{matrix} \begin{matrix} {E_{\infty} = {{\lambda_{\infty}c_{\infty}} - b_{\infty}}} \\ {= {{\frac{\lambda_{\infty}}{2\pi}{\int_{- \infty}^{+ \infty}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\mathbb{d}\omega}}}} - {\frac{1}{2\;\pi}{\int_{- \infty}^{+ \infty}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\mathbb{d}\omega}}}}}} \\ {> 0} \end{matrix} & (38) \end{matrix}$ represents the minimum transmit energy (threshold) required to avoid partitioning in the frequency domain. With E_(∞) as given by (38), we obtain SINR_(max) to be (use (28))

$\begin{matrix} {{{SINR}_{1}(\infty)} = {{a_{\infty} - \frac{c_{\infty}}{\lambda_{\infty}}} = {{a_{\infty} - \frac{c_{\infty}^{2}}{E_{\infty} + b_{\infty}}} > 0}}} & (39) \\ {and} & \; \\ \begin{matrix} {{{{F(\omega)}}^{2} = \frac{\sqrt{G_{n}(\omega)}\left( {{\lambda_{\infty}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)}},} & {{\omega } < {\infty.}} \end{matrix} & (40) \end{matrix}$ Clearly by further increasing the transmit energy in (39) beyond that in (38) we obtain

$\begin{matrix} {\left. {SINR}_{1}\rightarrow a_{\infty} \right. = {\frac{1}{2\pi}{\int_{- \infty}^{- \infty}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{{\mathbb{d}\omega}.}}}}} & (41) \end{matrix}$ It follows that to avoid any restrictions in the frequency domain for the transmit signal, the transmitter energy E must exceed a minimum threshold value E_(∞) given by (38) and (39) represents the maximum realizable SNR. By increasing E beyond E_(∞), the performance can be improved upto that in (41).

In general from (34) for a given desired bandwidth B_(o), the transmit energy E must exceed its threshold value E_(B) _(o) . With E>E_(B) _(o) and λ(E) as in (25), the corresponding optimum transmit signal transform is given by (see (21) (22))

$\begin{matrix} {{{F(\omega)}}^{2} = \left\{ \begin{matrix} {\frac{\sqrt{G_{n}(\omega)}\left( {{{\lambda(E)}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)},} & {\omega \in {\Omega_{+}\left( B_{o} \right)}} \\ {0,} & {\omega \in \Omega_{o}} \end{matrix} \right.} & (42) \end{matrix}$ and clearly this signal is different from the minimum threshold energy one in (36). From (28), the performance level SINR₁(E,B_(o)) corresponding to (42) is given by (35) with E_(B) _(o) replaced by E. Thus

$\begin{matrix} {{{SINR}_{1}\left( {E,B_{o}} \right)} = {{a_{o} - \frac{c_{o}^{2}}{E + b_{o}}} > {{{SINR}_{1}\left( B_{o} \right)}.}}} & (43) \end{matrix}$ From (43), for a given bandwidth B_(o), performance can be increased beyond that in (35) by increasing the transmit energy. Hence it follows that SINR₁(B_(o)) represents the minimum performance level for bandwidth B_(o) that is obtained by using its minimum threshold energy. It is quite possible that this improved performance SINR₁(E,B_(o)) can be equal to the minimum performance level corresponding to a higher bandwidth B₁>B_(o). This gives rise to the concept of Energy-Bandwidth tradeoff at a certain performance level. Undoubtedly this is quite useful when bandwidth is at premium.

FIGS. 5E-5F exhibit the transmit threshold energy and maximum output SINR₁(B_(o)) as a function of the desired bandwidth B_(o). Combining these figures using (35), an SINR vs. transmit threshold energy plot can be generated as in FIGS. 6A-C for each target situation.

For example, FIG. 6A-C corresponds to the three different target situations considered in FIG. 5 with clutter and noise spectra as shown there. Notice that each point on the SINR-Energy threshold curve for each target is associated with a specific desired bandwidth. Thus for bandwidth B₁, the minimum threshold energy required is E₁, and the corresponding SINR equals SINR₁(B₁) in (35). Let A represent the associated operating point in FIG. 6. Note that the operating point A corresponding to a bandwidth B₁ has different threshold energies and different performance levels for different targets. From (35), each operating point generates a distinct transmit waveform. As the bandwidth increases, from (39), SINR→SINR₁(∞).

Monotonic Property of SINR

The threshold energy and SINR associated with a higher bandwidth is higher. To prove this, consider two desired bandwidths B₁ and B₂ with B₂>B₁. Then from (32) we have

$\begin{matrix} {{\lambda_{2} = {{{\max\limits_{\omega \in B_{2}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} > \lambda_{1}} = {\max\limits_{\omega \in B_{1}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}}},} & (44) \end{matrix}$ and from FIG. 3, the corresponding operating bandwidths Ω₊(B₁) and Ω₊(B₂) satisfy Ω₊(B ₂)≧Ω₊(B ₁)  (45) From (34) (or (24)), the minimum threshold energies are given by

$\begin{matrix} \begin{matrix} {{E_{i} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{i})}}^{\;}{\sqrt{G_{n}(\omega)}\left( {\lambda_{i} - \frac{\sqrt{G_{n}(\omega)}}{Q(\omega)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{\mathbb{d}\omega}}}}},} & {{i = 1},2} \end{matrix} & (46) \end{matrix}$ and substituting (44) and (45) into (46) we get E₂>E₁  (47) Also from (27), the performance levels at threshold energy SINR₁(B_(i)) equals

$\begin{matrix} {{{SINR}_{1}\left( B_{i} \right)} = {\frac{1}{2\;\pi}{\int_{\Omega_{+}{(B_{i})}}^{\;}{\left( {{{Q(\omega)}} - \frac{\sqrt{G_{n}(\omega)}}{\lambda_{i}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}{\mathbb{d}\omega}}}}} & (48) \end{matrix}$ and an argument similar to (44)-(45) gives SINR₁(B ₂)≧SINR₁(B ₁)  (49) for B₂>B₁. Thus as FIGS. 5A-F-FIGS. 6A-C show, SINR₁(B_(i)) is a monotonically non -decreasing function of both bandwidth and energy. FIG. 7 illustrates this SINR-energy relation for the target with flat spectrum shown in FIG. 5A. In FIG. 7, the two operating points A and B are associated with bandwidths B₁ and B₂, threshold energies E₁ and E₂, and performance levels SINR₁(B₁) and SINR₁(B₂) respectively. Since B ₂ >B ₁

E ₂ ≧E ₁ and SINR₁(B ₂)≧SINR₁(B ₁).  (50) The distinct transmit waveforms |F₁(ω)|² and |F₂(ω)|² associated with these operating point A and B are given by (36) and they are shown in FIGS. 8A and 8B.

Consider the operating point A associated with the desired bandwidth B₁. If the transmit energy E is increased beyond the corresponding threshold value E₁ with bandwidth held constant at B₁, the performance SINR₁(E, B₁) increases beyond that at A since from (43)

$\begin{matrix} {{{SINR}_{1}\left( {E,B_{1}} \right)} = {{{a_{1} - \frac{c_{1}^{2}}{E + b_{1}}} \geq {a_{1} - \frac{c_{1}^{2}}{E_{1} + b_{1}}}} = {{SINR}_{1}\left( B_{1} \right)}}} & (51) \end{matrix}$ and it is upper bounded by a₁. Here a₁ corresponds to the SINR performance for bandwidth B₁ as the transmit energy E→∞. Note that a₁, b₁ and c₁ are the constants in (29)-(31) with Ω₊ replaced by Ω₊(B₁). The dashed curve Aa₁ in FIG. 7 represents SINR₁(E, B₁) for various values of E. From (42), each point on the curve Aa₁ generates a new transmit waveform as well.

Interestingly the dashed curves in FIG. 7 cannot cross over the optimum performance (solid) curve SINR(B_(i)). If not, assume the performance SINR₁(E,B₁) associated with the operating point A crosses over SINR(B_(i)) at some E₁′>E₁. Then from (47), there exists a frequency point B₁′>B₁ with threshold energy E₁′ and optimum performance SINR₁(B₁′). By assumption, SINR₁(E ₁ ′,B ₁)>SINR₁(B ₁′)  (52) But this is impossible since SINR₁(B₁′) corresponds to the maximum SINR realizable at bandwidth B₁′ with energy E₁′, and hence performance at a lower bandwidth B₁ with the same energy cannot exceed it. Hence (52) cannot be true and we must have SINR₁(E ₁ ′,B ₁)≦SINR₁(B ₁′),  (53) i.e., the curves Aa₁, Ba₂, etc. does not cross over the optimum performance curve ABD.

In FIG. 7, assume that the saturation performance value a ₁≧SINR₁(B ₂),  (54) i.e., the maximum performance level for bandwidth B₁ is greater than of equal to the performance level associated with the operating point B with a higher bandwidth B₂ and a higher threshold energy E₂. Draw a horizontal line through B to intersect the curve Aa₁ at C, and drop a perpendicular at C to intersect the x-axis at E₃. From (51) with E=E₃ we get SINR₁(E ₃ ,B ₁)=SINR₁(B ₂).  (55) Thus the operating point C on the curve Aa₁ is associated with energy E₃, bandwidth B₁ and corresponds to a performance level of SINR₁(B₂) associated with a higher bandwidth. Notice that E₃>E₂>E₁, and B₁<B₂.  (56) In other words, by increasing the transmit energy from E₁ to E₃ while holding the bandwidth constant at B₁, the performance equivalent to a higher bandwidth B₂ can be realized provided B₂ satisfies (54). As a result, energy-bandwidth tradeoff is possible within reasonable limits. The transmit waveform |F₃(ω)|² associated with the operating point C is obtained using (42) by replacing E and B₀ there with E₃ and B₁ respectively. and it is illustrated in FIG. 8C. In a similar manner, the waveforms corresponding to the operating points A and B in FIG. 7 can be obtained using equation (42) by replacing the energy-bandwidth pair (E,B₀) there with (E₁,B₂) and (E₂,B₂) respectively. These waveforms are shown in FIG. 8A and FIG. 8B respectively A comparison with FIGS. 8A and 8B show that the waveform at C is different from those associated with operating point A and B.

It is important to note that although the transmit waveform design |F₃(ω)|² and |F₁(ω)|² correspond to the same bandwidth (with different energies E₃ and E₁), one is not a scaled version of the other. Changing transmit energy from E₁ to E₃ unleashes the whole design procedure and ends up in a new waveform |F₃(ω)|² that maintains a performance level associated with a larger bandwidth B₂.

The question of how much bandwidth tradeoff can be achieved at an operating point is an interesting one. From the above argument, equality condition in (54) gives the upper bound on how much effective bandwidth increment can be achieved by increasing the transmit energy. Notice that for an operating point A, the desired bandwidth B₁ gives the operating bandwidth Ω₊(B₁) and from (29) the performance limit

$\begin{matrix} {a_{1} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{1})}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{\mathbb{d}\omega}}}}} & (57) \end{matrix}$ for bandwidth B₁ can be computed. Assume B₂>B₁, and from (48) SINR₁(B₂) the minimum performance at B₂ also can be computed, and for maximum bandwidth swapping the nonlinear equation a ₁=SINR₁(B ₂)  (58) must be solved for B₂. Then ΔB(B ₁)=B ₂ −B ₁  (59) represents the maximum bandwidth enhancement that can be realized at B₁. This is illustrated in for the target situation in FIG. 7. Notice that the maximum operating bandwidth if finite in any system due to sampling considerations and after normalization, it is represented by unity. Hence the upper limit in (59) must be min(1, B₂). This gives ΔB=min(1,B ₂)−B ₁  (60) and this explains the linear nature of ΔB for larger value of B_(i). In that case, bandwidth can be enhanced by 1−B₁ only.

The design approach described in this section requires the knowledge of the target characteristics in addition to the clutter and noise spectra.

A further embodiment of the present invention, which concerns savings in bandwidth and energy using waveform design is disclosed as follows.

The waveform design procedure described in the previous section can be extended in such a way as to result in simultaneous savings in bandwidth and energy with respect to any prior art transmit waveform such as a chirp waveform.

To realize this goal, let F_(o)(ω) represent the transform of any conventional waveform such as a chirp signal with energy E_(o) and using bandwidth B_(o). Then

$\begin{matrix} {E_{o} = {\frac{1}{2\pi}{\int_{- B_{o}}^{B_{o}}{{{F_{o}(\omega)}}^{2}{\mathbb{d}\omega}}}}} & (61) \end{matrix}$ and for a given target with transform Q(ω), and clutter and noise with spectra G_(c)(ω) and G_(n)(ω), from equation (12), the optimum receiver is given by

$\begin{matrix} {{H_{o}(\omega)} = {\frac{{Q^{*}(\omega)}{F_{o}^{*}(\omega)}}{{{G_{c}(\omega)}{{F_{o}(\omega)}}^{2}} + {G_{n}(\omega)}}{\mathbb{e}}^{{- j}\;\omega\; t_{o}}}} & (62) \end{matrix}$ and from equation (11), the corresponding maximum SINR output is given by

$\begin{matrix} {{SINR}_{o} = {\frac{1}{2\pi}{\int_{- B_{o}}^{B_{o}}{\frac{{{Q(\omega)}}^{2}{{F_{o}(\omega)}}^{2}}{{{G_{c}(\omega)}{{F_{o}(\omega)}}^{2}} + {G_{n}(\omega)}}{{\mathbb{d}\omega}.}}}}} & (63) \end{matrix}$ Notice that the output SINR_(o) in equation (63) is a function of two independent parameters, the desired bandwidth B_(o) and energy E_(o), and for comparison purposes these parameters can be made equal to their corresponding values obtained using the optimum waveform design as described below.

In the case of the optimum waveform design, for a given desired bandwidth B_(o), the minimum required energy is given by E_(B) _(o) in equation (34), and the corresponding maximum output SINR equals SINR₁ and that is given by

$\begin{matrix} {{SINR}_{1} = {{{SINR}_{1}\left( B_{o} \right)} = {a_{o} - \frac{c_{o}^{2}}{E_{o} + b_{o}}}}} & (64) \end{matrix}$ as in equation (35), where E_(o)=E_(B) _(o)   (65) and a_(o), b_(o), c_(o) are as described in equations (29)-(31) with Ω₊ there replaced by the operating bandwidth Ω₊(B_(o)) as defined in equation (33).

From the previous section on “An Embodiment of the Present Invention—The Desired Band Approach”, to compute the SINR₁ in equation (64) corresponding to the optimum method, the desired frequency bandwidth B_(o) is used first to define an operating bandwidth Ω₊(B_(o)), and then integration is carried out over Ω₊(B_(o)) in equations (27)-(34). As FIG. 3B shows the operating bandwidth Ω₊(B_(o)) in some cases can exceed the desired band (−B_(o),B_(o)). In the context of bandwidth comparison with respect to a given waveform F_(o)(ω) with bandwidth B_(o), it is necessary to make sure that the operating bandwidth Ω₊(B_(o)) does not exceed the desired bandwidth (−B_(o),B_(o)). To realize this goal, a transmitter output filter P₁(ω) as in FIG. 11 whose pass-band bandwidth B coincides with the given desired bandwidth can be employed. From FIG. 1, the transmitter output filter P₁(ω) modifies the given target transform Q(ω) to Q(ω)P₁(ω), and also modifies the given clutter spectrum G_(c)(ω) to G_(c)(ω)P₁(ω). As a result, when FIG. 3B is redrawn using this modified target transform Q(ω)P₁(ω) to determine the operating bandwidth Ω₊(B_(o)), it is easy to see that Ω₊(B_(o)) coincides with (−B_(o),B_(o)) as shown in FIG. 12D, and hence the operating bandwidth never exceeds the desired bandwidth B in this approach. This technique of inserting a transmitter output filter P₁(ω) with pass-band with B_(o) is employed whenever bandwidth savings comparison with respect to a given waveform is being investigated. Here onwards we will assume that the operating bandwidth Ω₊(B_(o)) and the desired band (−B_(o),B_(o)) are equal to each other.

The solid curve 1001 in FIG. 10 represents the optimum SINR₁ in equation (64) as a function of energy E_(o) for a hypothetical target, noise and clutter spectra using the optimum waveform design in equation (36) with bandwidth B_(o). By employing a transmitter output filter P₁(ω) with pass-band with B_(o), the operating bandwidth Ω₊(B_(o)) here coincides with the desired bandwidth B_(o). Thus each point on the solid curve 1001 corresponds to an energy-bandwidth pair (E_(o), B_(o)). Using the same pair (E_(o), B_(o)) for the given transmit signal such as the chirp waveform, the corresponding SINR_(o) is computed using equation (63) and it is plotted as the solid curve 1002 in FIG. 10. The given waveform F_(o)(ω) being generally suboptimum, the SINR_(o) curve 1002 will usually fall below the optimum SINR₁ curve 1001 as shown in FIG. 10. Thus points A and B in FIG. 10 (on curves 1001 and 1002 respectively) correspond to the same bandwidth-energy pair (E_(o), B_(o)) but with different performance levels SINR_(o) and SINR₁ respectively with SINR_(o)≦SINR₁  (66) where SINR_(o) represents the performance in equation (63) when the given transmit waveform is used, and SINR₁ represents the performance in equation (64) when the optimum transmit waveform given by equation (36) is used. Draw a horizontal line 1003 through A to meet the solid curve 1001 at C as shown in FIG. 10. The operating point C in 1001 has performance level equal to SINR_(o) and corresponds to some energy -bandwidth pair (E_(C), B_(C)) that can be solved by equating SINR₁(B_(C)) with SINR_(o). Thus B_(C) is obtained by solving

$\begin{matrix} {{{SINR}_{1}\left( B_{C} \right)} = {{a_{C} - \frac{c_{C}^{2}}{E_{C} + b_{C}}} = {{SINR}_{o}.}}} & (67) \end{matrix}$

Here a_(C), b_(C), c_(C) are as described in equations (29)-(31) with Ω₊ replaced by the operating bandwidth Ω₊(B_(C))=B_(C), and E_(C) is given by equation (34) with B_(o) replaced by B_(C). Since SINR_(o)≦SINR₁(B_(o)), and the operating point C falls below the point B in FIG. 10, from equation (50) we have B_(C)<B_(o), E_(C)<E_(o).  (68) As a result, the optimum waveform design F₁(ω) corresponding to the operating point C is given by equation (36) with B_(o) replaced by B_(C) with the pass-band for P₁(ω) in FIG. 11 extending up to B_(C). Hence F₁(ω) uses the energy-bandwidth pair (E_(C), B_(C)) and from equation (68) this pair is uniformly superior to the given pair (E_(o), B_(o)), and it performs at the same level SINR_(o) as the given transmit waveform. Thus for a given waveform F_(o)(ω) associated with operating point A in curve 1002, simultaneous savings in both bandwidth and energy are possible by making use of the optimum waveform design F_(o)(ω) that corresponds to the operating point C in curve 1001. The operating point C corresponds to the minimum energy design for the same performance level and the maximum saving in energy equals E_(o)−E_(C)>0, and the corresponding bandwidth saving equals B_(o)−B_(C).

Interestingly, additional savings in bandwidth can be made at the expense of more transmit energy. To see this, consider an operating point D that is below C on the optimum design curve 101 marked SINR₁(B_(i)). From equation (50), the design point D operates with lower bandwidth B_(D)<B_(C) and uses energy E_(D) given by equation (34) with B_(o) replaced by B_(D), and provides lower performance level SINR₁(B_(D)). To increase the performance level beyond SINR₁(B_(D)), the transmit energy E can be increased beyond its threshold value E_(D) and from equation (43) and equation (51) increase in transmit energy gives the performance

$\begin{matrix} {{{SINR}_{1}\left( {E,B_{D}} \right)} = {a_{D} - \frac{c_{D}^{2}}{E + b_{D}}}} & (69) \end{matrix}$ where a_(D), b_(D) and c_(D) are as in equations (29), (30) and (31) with Ω₊ replaced by Ω₊(B_(D))=B_(D). The dashed curve Da_(D) (curve 1004) in FIG. 10 represents SINR₁(E, B_(D)) for various values of energy E. Each point on curve 1004 generates a new transmit waveform as given by equation (42). The dashed curve 1004 meets the horizontal line AC (curve 1003) at F that corresponds to energy E_(F). Hence from equation (69) SINR₁(E _(F) ,B _(D))=SINR_(o).  (70) In other words, the optimum design |F₂(ω)|² in equation (42) with energy-bandwidth pair (E_(F), B_(D)) also maintains the same output performance as SINR_(o). In this case, the useable bandwidth has further decreased from B_(C) to B_(D), at the expense of increase in transmit energy from E_(C) to E_(F).

Proceeding in this manner, we observe that the dashed curve 1005 in FIG. 10 given by Ga_(G) that passes through the initial design point A uses minimum bandwidth B_(G). The operating point G on the optimum SINR₁ curve 1001 uses bandwidth B_(G), energy E_(G) and gives out performance SINR₁(B_(G)). By increasing energy from E_(G) to E_(o), while bandwidth is held at B_(G), the performance SINR₁(E, B_(G)) defined as in equation (69) with B_(D) replaced by B_(G), can be increased to SINR_(o), and proceeding as in equation (70) by equating SINR₁(E, B_(G)) to SINR_(o) we get SINR₁(E _(o) ,B _(G))=SINR_(o),  (71) i.e., the energy required for the optimum design at point A on curve 1005 is the same as the original transmit signal energy E_(o). Thus B_(G) represents the minimum bandwidth that can be used without the transmit signal energy exceeding E_(o) for the optimum design, and B_(o)−B_(G) represents the corresponding maximum savings in bandwidth. Notice that the design point A on the dashed curve 1005 generates a new transmit waveform |F₃(ω)|² according to equation (42) with B_(o) replaced by B_(G) there.

If we are prepared to use additional transmit energy beyond E_(o), then by starting with operating points that are below G on the optimum curve 1001 and using additional transmit energy E>E_(o), performance equal to SINR_(o) can be realized. Since operating points below G have lower bandwidth compared to B_(G), these designs use even lower bandwidths at the expense of additional transmit energy.

In summary, optimum designs that start from operating points such as D located in between G and C on the curve 1001, result in simultaneous savings in energy and bandwidth for the same SINR performance compared to the given transmit waveform such as at point A on curve 1002.

To validate the simultaneous savings in bandwidth and energy using waveform diversity, FIG. 13A and FIG. 13B show the actual SINR-Energy plots employing optimum waveform design (curve 1301) as well as a given chirp transmit signal (curve 1302). Both methods use a target transform Q(ω) as in FIG. 12A, and noise and clutter spectra G_(n)(ω) and G_(c)(ω) as in FIG. 12B and FIG. 12C, respectively. As described earlier the target and clutter has been modified using the transmitter output filter P₁(ω) in FIG. 11 so that at every stage the operating band Ω₊(B_(i)) coincides with the desired bandwidth B_(i). This is illustrated in FIG. 12D for a typical bandwidth B_(i). FIG. 13B shows an enlarged version for FIG. 13A with the features around the points of interest A, B, C, D, F and G magnified.

The given chirp waveform F_(o)(ω) operating at point A in FIG. 13 uses the energy-bandwidth pair (E_(o), B_(o))=(8.88 dB, 0.15) with output performance given by SINR_(o)=6.15 dB. All bandwidths here are expressed as a fraction of the maximum available bandwidth dictated by the sampling period. The operating point C using waveform diversity on the optimum SINR₁ curve 1301 with performance equal to SINR_(o), uses the energy-bandwidth pair (E_(C), B_(C))=(5.875 dB, 0.1193) resulting in simultaneous savings in both transmit energy and bandwidth compared to the chirp transmit waveform. Notice that compared to the given chirp waveform, the bandwidth requirement for the optimum waveform |F₁(ω)|² given by equation (36) with B_(o)=B_(C) has been reduced from 0.15 to 0.1193, whereas the energy requirement has been reduced from 8.88 dB to 5.875 dB without sacrificing the performance.

The operating point F also corresponds to the same performance level as SINR_(o), and it lies on curve 1304 that corresponds to SINR₁(E, B_(D)) curve Da_(D) with B_(D)=0.065. The performance SINR_(o) is met here with energy E_(F)=7.01 dB. In other words, the optimum waveform |F₂(ω)|² given by equation (42) with E=E_(F) and B_(o)=B_(D) gives the same performance SINR_(o) as the given chirp transmit waveform, but uses less bandwidth and less energy. In this case, bandwidth requirement is reduced from 0.15 to 0.065, and the energy requirement is reduced from 8.88 dB to 7.01 dB. Notice that compared to the optimum waveform |F₁(ω)|² at C, the waveform |F₂(ω)|² at point F uses less bandwidth, but more energy. These waveform transform magnitude functions are plotted in FIG. 14B and FIG. 14C respectively along with the given transmit waveform transform magnitude |F_(o)(ω)|² plotted in FIG. 14A.

Finally the curve 1305 that starts at G passes through A and the optimum waveform |F₃(ω)|² given by equation (42) with E=E_(o) and B_(o)=B_(G)=0.0589 also maintains the same performance SINR_(o). Notice that |F₄(ω)|² design plotted in FIG. 14D uses the least amount of bandwidth (about ⅓ bandwidth compared to the given chirp waveform) without the transmit energy exceeding the initial value E_(o).

Thus, the design procedure illustrated in FIG. 10 and the results of actual computation in FIGS. 13-FIG. 14 using data from FIG. 11 and FIG. 12 show that simultaneous savings in bandwidth and energy are indeed possible using optimum waveform design compared to a given transmit waveform.

Finally to convert the optimum Fourier transform magnitude information in FIGS. 14A, B, C and D into a practically useful time domain signal, a suitable phase function ψ_(i)(ω) needs to be determined in each case. Let A(ω) represent the given Fourier transform magnitude |F_(i)(ω)| in FIGS. 14B, C and D, and together with the standard inverse Fourier transform operation f_(i)(t)

|F_(i)(ω)|e^(−jψ) ^(i) ^((ω))  (72) gives the desired time domain signal f_(i)(t), i=1, 2, 3, 4. Here

 represents the Fourier transform operation. The phase functions ψ_(i)(ω) are in general arbitrary in equation (72) and the freedom present in their selection can be used to “shape” the final time -domain signal f_(i)(t) in equation (72) to be practically useful. One useful feature for time domain transmit signals is the constant envelope property that maximizes the transmit power efficiency. In that case, the final time domain waveform f_(i)(t) will ideally have constant envelope or in practice approximately possess constant envelope thereby distributing the peak transmit power more evenly.

To achieve this goal, the method of alternating convex projections can be used such as described in D. C. Youla, “Mathematical Theory of image Restoration by the Method of Convex projections,” Chapter 2, Theorem 2.4-1, in Image Recovery: Theory and Application edited by Henry Stark, pages 29-77, Academic press, Inc., New York, 1987. The desired signal f_(i)(t) should maintain the band-limited property, as well as satisfy the given Fourier transform magnitude information A(ω) within the respective bandwidth as exhibited in FIG. 14. In addition, the desired signal also should maintain the constant envelope property in time domain. Although both the band-limited property and the constant envelope property do form convex sets, signals with the same Fourier transform magnitude do not form a convex set. Convex sets have the interesting property that starting with any given signal, projection operators associated with the convex sets determine a unique nearest neighbor to the given signal that is within the convex set. Although the magnitude substitution operation does not enjoy this unique nearest neighbor property, when iterated together with a projection operator it possesses an error reduction property at every stage, and the combined operator has been seen to converge in simulation studies, such as shown in A. Levi and H. Stark, “Image Restoration by the Method of Generalized Projections with Applications to Restoration from Magnitude,” IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP'84, pages 88-91, March 1984 (See Theorem I on page 88 of Levi and Stark). To exploit these features, let P_(B) _(o) represent the combined band -limiting operation onto the convex set C_(B) _(o) that is band-limited to (−B_(o), B_(o)) as well as the magnitude substitution operation using the given Fourier transform magnitude A(ω) from FIG. 14. Similarly let P_(c) represent the projection operator onto the convex set C_(c) that is envelope limited to ±c. Then for any arbitrary signal f(t)

F(ω)=|F(ω)|e ^(−jψ(ω)),  (73) the combined projection operator and the magnitude substitution operation P_(B) _(o) is defined in terms of its transform as

$\begin{matrix} \left. {P_{B_{o}}\left( {f(t)} \right)}\leftrightarrow\left\{ {\begin{matrix} {{{A(\omega)}{\mathbb{e}}^{- {{j\psi}{(\omega)}}}},} & {{\omega } \leq B_{o}} \\ {0,} & {{\omega } > B_{o}} \end{matrix},} \right. \right. & (74) \end{matrix}$ where A(ω) represents the given Fourier transform magnitude within the desired bandwidth (−B_(o), B_(o)). Similarly

$\begin{matrix} {{P_{c}\left( {f(t)} \right)} = \left\{ \begin{matrix} {{f(t)},} & {{{f(t)}} \leq c} \\ {{+ c},} & {{f(t)} > c} \\ {{- c},} & {{f(t)} < {- c}} \end{matrix} \right.} & (75) \end{matrix}$ represents the projection operator onto the constant envelope convex set, where c represents a suitable amplitude level to be determined that acts as the upper limit on the envelope. Starting with an arbitrary phase ψ_(o)(ω) and the given bandlimited magnitude transform A(ω), one may define f _(o)(t)

F _(o)(ω)=A(ω)e ^(−jψ) ^(o) ^((ω))  (76) as the initial signal. The method of alternating projection applies repeatedly the given projection operators in an alternating fashion and the combined iterate is known to converge weakly to a point in the intersection of the respective convex sets as shown in D. C. Youla and H. Webb, “Image Restoration by the Method of Convex Projections,” IEEE Transactions on Medical imaging, Vol. MI-1, October 1982. Following that approach, in the present situation, the operator P_(B) _(o) and the projection operator P_(c) defined in equations (74)-(75) are applied to the initial signal f_(o)(t) in an alternating fashion. In this case with P=P_(c)P_(B) _(o) ,  (77) representing the sequentially combined operator, we have f _(n+1)(t)=P _(c) P _(B) _(o) f _(n)(t)=Pf _(n)(t), n=0,1,2 . . .  (78) and the n^(th) iteration f _(n)(t)=P ^(n) f _(o)(t)  (79) will have both the desired properties. As remarked earlier, in the case of purely projection operators, the iterate in equation (79) converges weakly to a point in the intersection set of the convex sets C_(B) _(o) and C_(c). In the present case although P_(c) is a projection operator, P_(B) _(o) is not a projection operator and hence the convergence of the iterate in equations (78)-(79) has been determined only experimentally. This conclusion is also supported by the error reduction property ∥f _(n+1)(t)−P _(B) _(o) f _(n+1)(t)∥≦∥f _(n)(t)−P _(B) _(o) f _(n)(t)∥  (80) mentioned earlier. Finally, the constant amplitude level c in equation (75) can be determined by scaling the time domain waveform f_(n+1)(t) so as to maintain its prescribed energy level. The number of iterations to be employed in equations (78)-(79) can be adjusted by stipulating the error function in equation (80) to be within an acceptable level.

FIG. 15A shows the original chirp waveform that is constant in magnitude to start with. FIGS. 15B, C and D show the time-domain signals obtained iteratively in the manner described above as in equations (76)-(80) starting with FIGS. 14B, 14C and 14D respectively. Notice that the waveforms in FIGS. 15B, 15C and 15D have approximately constant envelope, and uses smaller bandwidth and smaller energy levels compared to the original chirp signal, and they all perform at the same level.

In summary, the design procedure illustrated in FIG. 10 and the results of actual computation in FIG. 13-FIG. 15 using data from FIG. 11 and FIG. 12 show that simultaneous savings in bandwidth and energy are indeed possible using optimum waveform design compared to a given transmit waveform. The magnitude information so obtained can be trained to generate an almost constant envelope signal as in equations (78)-(80), and scaled properly to maintain the prescribed energy level in each case. FIG. 16 shows the computer processor 1610 that implements one or more methods in accordance with embodiments of the present invention described above by receiving at its input 1610 a appropriate design parameters and outputting a desired transmit signal.

Although the invention has been described by reference to particular illustrative embodiments thereof, many changes and modifications of the invention may become apparent to those skilled in the art without departing from the spirit and scope of the invention. It is therefore intended to include within this patent all such changes and modifications as may reasonably and properly be included within the scope of the present invention's contribution to the art. 

1. A method comprising providing a transmitter and a receiver; outputting a transmit signal f_(o)(t) from the transmitter towards a target and towards interference; wherein the target produces a target signal; wherein the transmit signal f_(o)(t) has a transmit signal bandwidth B_(o), transmit signal energy, and a transmit signal waveform; and further comprising receiving a combination signal at the receiver, wherein the combination signal includes noise and the transmit signal f_(o)(t) modified by interacting with the target and the interference; wherein the receiver has a filter having a transfer function of ${H_{o}(\omega)} = {\frac{{Q^{*}(\omega)}{F_{o}^{*}(\omega)}}{{{G_{c}(\omega)}{{F_{o}(\omega)}}^{2}} + {G_{n}(\omega)}}{\mathbb{e}}^{{- {j\omega}}\; t_{o}}}$ and the filter acts on the combination signal to form a receiver output signal having a receiver output signal waveform; wherein F_(o)(ω) represents the Fourier transform of the transmit signal f_(o)(t); wherein Q(ω) is the target signal Fourier transform; G_(c)(ω) is the interference spectrum; and G_(n)(ω) is the noise spectrum; wherein the receiver output signal has a receiver output signal waveform that describes an output signal to interference to noise ratio (SINR) performance; and reducing both the transmit signal bandwidth and transmit signal energy simultaneously by modifying the transmit signal waveform and receiver output signal waveforms without sacrificing the output SINR performance level.
 2. The method of claim 1 further comprising selecting an initial desired bandwidth B_(o) for the transmit signal bandwidth and determining the required energy E_(o) for the transmit signal according to $E_{o} = {\frac{1}{2\pi}{\int_{- B_{0}}^{B_{0}}{\sqrt{G_{n}(\omega)}\left( {\lambda_{o} - \frac{\sqrt{G_{n}(\omega)}}{Q(\omega)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}$ wherein ${\lambda_{o} = {\max\limits_{\omega \in B_{o}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}},$ and scaling the transmit signal so that its energy is made equal to E_(o) above; and determining the performance for the transmit signal F_(o)(ω) with transmit energy E_(o) and bandwidth B_(o), receiver signal H_(o)(ω) above is given by ${SINR}_{o} = {\frac{1}{2\pi}{\int_{- B_{o}}^{B_{o}}{\frac{{{Q(\omega)}}^{2}{{F_{o}(\omega)}}^{2}}{{{G_{c}(\omega)}{{F_{o}(\omega)}}^{2}} + {G_{n}(\omega)}}\ {{\mathbb{d}\omega}.}}}}$ wherein Q(ω) is the target signal Fourier transform; G_(c)(ω) is the interference spectrum; and G_(n)(ω) is the noise spectrum.
 3. The method of claim 1 further comprising selecting a second bandwidth B_(C) smaller than the given bandwidth B_(o) so as to satisfy the condition ${{{SINR}_{1}\left( {E_{c},B_{c}} \right)} = {{a_{c} - \frac{c_{c}^{2}}{E_{c} + b_{c}}} = {SINR}_{o}}},$ where the second energy level E_(c) smaller than the energy E_(o) is given by $E_{c} = {\frac{1}{2\pi}{\int_{- B_{C}}^{B_{C}}{\sqrt{G_{n}(\omega)}\left( {\lambda_{c} - \frac{\sqrt{G_{n}(\omega)}}{Q(\omega)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}$ with ${\lambda_{c} = {\max\limits_{\omega \in B_{C}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}},{and}$ ${a_{c} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{C})}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},{b_{c} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{C})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},{and}$ ${c_{c} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{C})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},$ where Ω₊(B_(C)) represents the frequency band (−B_(C)≦ω≦B_(C)).
 4. The method of claim 3 further comprising selecting a third bandwidth B_(D) smaller than the second bandwidth B_(C) and solving for a third energy E_(F) so as to satisfy the condition ${{{SINR}_{1}\left( {E_{F},B_{D}} \right)} = {{a_{D} - \frac{c_{D}^{2}}{E_{F} + b_{D}}} = {SINR}_{o}}},{and}$ ${a_{D} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{D})}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},{b_{D} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{D})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},{and}$ ${c_{D} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{D})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},$ where Ω₊(B_(D)) represents the frequency band (−B_(D)≦ω≦B_(D)).
 5. The method of claim 4 further comprising constructing a third transmit signal having a third transmit signal waveform; wherein the third transmit signal is different from the first and second transmit signals and the third transmit signal waveform is different from the first and second transmit signal waveforms; wherein the third transmit signal has a prescribed bandwidth which is B_(D) that is smaller than the second transmit signal bandwidth B_(C); wherein the third transmit signal has an energy which is E_(F) that is greater than the second transmit signal energy E_(C), but smaller than the first transmit signal energy E_(o); wherein the third transmit signal has Fourier transform F₂(ω) whose magnitude function is given by ${{F_{2}(\omega)}}^{2} = \left\{ {{\begin{matrix} {\frac{\sqrt{G_{n}(\omega)}\left( {{{\lambda\left( E_{F} \right)}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)},} & {\omega \in {\Omega_{+}\left( B_{D} \right)}} \\ {0,} & {otherwise} \end{matrix}{wherein}{\lambda\left( E_{F} \right)}} = {\frac{E_{F} + {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{D})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}}{\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{D})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}.}} \right.$ and wherein the third, second and first transmit signals have the same performance index SINR_(o) in terms of target detection in interference and noise, when the third signal is used in conjunction with a receiver filter that has a Fourier transform H_(opt)(ω) given by ${H_{opt}(\omega)} = {\frac{{Q^{*}(\omega)}{F_{2}^{*}(\omega)}}{{{G_{c}(\omega)}{{F_{2}(\omega)}}^{2}} + {G_{n}(\omega)}}{\mathbb{e}}^{{- {j\omega}}\; t_{o}}}$ wherein t_(o) is a decision instant at which the target signal is to be detected.
 6. The method of claim 5 further comprising generating a time domain signal that is constant envelope substantially everywhere while maintaining the given third transform magnitude |F₂(ω)| by starting with any suitable signal f_(o)(t) and iteratively computing f _(n+1)(t)=P _(c) {P _(B) _(D) (f _(n)(t))}, n=0,1,2 . . . wherein $\left. {P_{B_{D}}\left( {f_{n}(t)} \right)}\leftrightarrow\left\{ {\begin{matrix} {{{{F_{2}(\omega)}}{\mathbb{e}}^{- {{j\psi}_{n}{(\omega)}}}},} & {{\omega } \leq B_{D}} \\ {0,} & {{\omega } > B_{D}} \end{matrix},} \right. \right.$ with ⇄ representing the standard Fourier transform operation, ψ_(n)(ω) the phase function of the Fourier transform of f_(n)(t), and ${P_{c}\left\{ {f(t)} \right\}} = \left\{ {\begin{matrix} {{f(t)},} & {{{f(t)}} \leq c} \\ {{+ c},} & {{f(t)} > c} \\ {{- c},} & {{f(t)} < {- c}} \end{matrix},} \right.$ and determining the constant envelope level c above by normalizing the final iterated signal f_(n+1)(t) to maintain the prescribed energy level E_(F) wherein normalizing the final iterated signal F_(n+1)(t) gives the third transmit signal waveform and wherein the third transmit signal waveform maintain constant envelope substantially everywhere.
 7. The method of claim 3 further comprising selecting a fourth bandwidth B_(G) smaller than the third bandwidth B_(D) so as to satisfy the condition ${{{SINR}_{1}\left( {E_{o},B_{G}} \right)} = {{a_{G} - \frac{c_{G}^{2}}{E_{o} + b_{G}}} = {SINR}_{o}}},$ where E_(o) corresponds to the prescribed energy, and ${a_{G} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{G})}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},{b_{G} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{G})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},{and}$ ${c_{G} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{G})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},$ where Ω₊(B_(G)) represents the frequency band (−B_(G)≦ω≦B_(G)).
 8. The method of claim 7 further comprising constructing a fourth transmit signal having a fourth transmit signal waveform; wherein the fourth transmit signal is different from the first, second and third transmit signals and the fourth transmit signal waveform is different from the first, second and third transmit signal waveforms; wherein the fourth transmit signal has a bandwidth which is B_(G) that is smaller than the third transmit signal bandwidth B_(D); wherein the fourth transmit signal has energy E_(o) that is the same as the first transmit signal energy; wherein the fourth transmit signal has transform F₃(ω) whose magnitude function is given by ${{F_{3}(\omega)}}^{2} = \left\{ {{\begin{matrix} {\frac{\sqrt{G_{n}(\omega)}\left( {{{\lambda\left( E_{o} \right)}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)},} & {\omega \in {\Omega_{+}\left( B_{G} \right)}} \\ {0,} & {otherwise} \end{matrix}{wherein}{\lambda\left( E_{o} \right)}} = {\frac{E_{o} + {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{G})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}}{\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{G})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}.}} \right.$ and wherein the fourth, third, second and first transmit signals have the same performance index SINR_(o) in terms of target detection in interference and noise, when used in conjunction with a receiver filter for the receiver such that the receiver filter has a Fourier transform H_(opt)(ω) given by ${H_{opt}(\omega)} = {\frac{{Q^{*}(\omega)}{F_{3}^{*}(\omega)}}{{{G_{c}(\omega)}{{F_{3}(\omega)}}^{2}} + {G_{n}(\omega)}}{\mathbb{e}}^{{- {j\omega}}\; t_{o}}}$ wherein t_(o) is a decision instant at which the target signal is to be detected.
 9. The method of claim 8 further comprising generating a time domain signal that is constant envelope substantially everywhere while maintaining the given fourth transform magnitude |F₃(ω)| by starting with any suitable signal f_(o)(t) and iteratively computing f _(n+1)(t)=P _(c) {P _(B) _(G) (f _(n)(t))}, n=0,1,2 . . . wherein $\left. {P_{B_{G}}\left( {f_{n}(t)} \right)}\leftrightarrow\left\{ {\begin{matrix} {{{{F_{3}(\omega)}}{\mathbb{e}}^{- {{j\psi}_{n}{(\omega)}}}},} & {{\omega } \leq B_{G}} \\ {0,} & {{\omega } > B_{G}} \end{matrix},} \right. \right.$ with ⇄ representing the standard Fourier transform operation, ψ_(n)(ω) the phase function of the Fourier transform of f_(n)(t), and ${P_{c}\left\{ {f(t)} \right\}} = \left\{ {\begin{matrix} {{f(t)},} & {{{f(t)}} \leq c} \\ {{+ c},} & {{f(t)} > c} \\ {{- c},} & {{f(t)} < {- c}} \end{matrix},} \right.$ and determining the constant envelope level c above by normalizing the final iterated signal f_(n+1)(t) to maintain the prescribed energy level E_(o)wherein normalizing the final iterated signal F_(n+1)(t) gives the fourth transmit signal waveform;and wherein the fourth transmit signal waveform maintains constant envelope substantially everywhere.
 10. A method comprising constructing a first transmit signal having a first transmit signal waveform; constructing a second transmit signal having a second transmit signal waveform; wherein the first transmit signal is different from the second transmit signal and the first transmit signal waveform is different from the second transmit signal waveform; wherein the first transmit signal has a prescribed bandwidth which is B_(o); wherein the first transmit signal has a prescribed energy which is E_(o); wherein the second transmit signal has a smaller bandwidth which is B_(C); wherein the second transmit signal has a smaller energy level which is E_(c); wherein the first transmit signal has a transform F_(o)(ω) which is user given; wherein the second transmit signal has a transform F₁(ω) whose magnitude function is given by ${{F_{1}(\omega)}}^{2} = \left\{ {\begin{matrix} {{\sqrt{G_{n}(\omega)}\left( {\left( {\max\limits_{\omega \in B_{C}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right) - \frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}},} & {{- B_{c}} \leq \omega \leq B_{c}} \\ {0,} & {otherwise} \end{matrix};} \right.$ and wherein the first and second transmit signals have the same performance index SINR_(o) given by ${SINR}_{o} = {\frac{1}{2\pi}{\int_{- B_{o}}^{B_{o}}{\frac{{{Q(\omega)}}^{2}{{F_{o}(\omega)}}^{2}}{{{G_{c}(\omega)}{{F_{o}(\omega)}}^{2}} + {G_{n}(\omega)}}\ {\mathbb{d}\omega}}}}$ in terms of target detection in interference and noise, when used in conjunction with a receiver filter for a receiver such that the receiver filter has a Fourier transform H_(opt)(ω) given by ${H_{opt}(\omega)} = {\frac{{Q^{*}(\omega)}{F^{*}(\omega)}}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}{\mathbb{e}}^{{- {j\omega}}\; t_{o}}}$ wherein Q(ω) is the target signal Fourier transform; G_(c)(ω) is the interference spectrum; and G_(n)(ω) is the noise spectrum and wherein t_(o) is a decision instant at which the target signal is to be detected, and these two waveforms F_(o)(ω) and F₁(ω) substituted for F(ω) above accordingly;and further comprising outputting the first transmit signal from a transmitter and towards a target; outputting the second transmit signal from the transmitter and towards the target; receiving one or morn received signals at a receiver as a result of the outputting of the first transmit signal and the outputting of the second transmit signal.
 11. The method of claim 10 further comprising generating a time domain signal that is constant envelope substantially everywhere while maintaining the given second transform magnitude |F₁(ω)|, by starting with any signal f_(o)(t) and iteratively computing f _(n+1)(t)=P _(c) {P _(B) _(c) (f _(n)(t))}, n=0,1,2 . . . where ${P_{B_{c}}\left( {f_{n}(t)} \right)}\left\{ {\begin{matrix} {{{{F_{1}(\omega)}}{\mathbb{e}}^{- {{j\psi}_{n}{(\omega)}}}},} & {{\omega } \leq B_{c}} \\ {0,} & {{\omega } > B_{c}} \end{matrix},} \right.$ with ⇄ representing the standard Fourier transform operation, ψ_(n)(ω) the phase function of the Fourier transform of f_(n)(t), and ${P_{c}\left\{ {f(t)} \right\}} = \left\{ {\begin{matrix} {{f(t)},} & {{{f(t)}} \leq c} \\ {{+ c},} & {{f(t)} > c} \\ {{- c},} & {{f(t)} < {- c}} \end{matrix},} \right.$ and determining the constant envelope level c above by normalizing the final iterated signal f_(n+1)(t) to maintain the prescribed energy level E_(c) and wherein normalizing the final iterated signal f_(n+1)(t) gives the second transmit signal waveform wherein the second transmit signal waveform maintains constant envelope substantially everywhere.
 12. An apparatus comprising a transmitter; a receiver; and a computer processor; wherein the transmitter is configured to output a transmit signal f_(o)(t) from the transmitter towards a target and towards interference; wherein the target produces a target signal; wherein the transmit signal f_(o)(t) has a transmit signal bandwidth B_(o), transmit signal energy, and a transmit signal waveform; and wherein the receiver is configured to receive a combination signal at the receiver, wherein the combination signal includes noise and the transmit signal f_(o)(t) modified by interacting with the target and the interference; wherein the receiver has a filter having a transfer function of ${H_{o}(\omega)} = {\frac{{Q^{*}(\omega)}{F_{o}^{*}(\omega)}}{{G_{c}(\omega)}{{{F_{o}(\omega)}^{2} + {G_{n}(\omega)}}}}{\mathbb{e}}^{- {j\omega t}_{o}}}$ and the filter acts on the combination signal to form a receiver output signal having a receiver output signal waveform; wherein F_(o)(ω) represents the Fourier transform of the transmit signal f_(o)(t); wherein Q(ω) is the target signal Fourier transform; G_(c)(ω) is the interference spectrum; and G_(n)(ω) is the noise spectrum; wherein the receiver output signal has a receiver output signal waveform that describes an output signal to interference to noise ratio (SINR) performance; and wherein the computer processor is programmed to reduce both the transmit signal bandwidth and transmit signal energy simultaneously by causing the transmit signal waveform and the receiver output signal waveform to be modified without sacrificing the output SINR performance level.
 13. The apparatus of claim 12 wherein the computer processor programmed to select an initial desired bandwidth B_(o) for the transmit signal bandwidth and the computer processor is programmed to determine the required energy E_(o) for the transmit signal according to $E_{o} = {\frac{1}{2\pi}{\int_{- B_{0}}^{B_{0}}{\sqrt{G_{n}(\omega)}\left( {\lambda_{o} - \frac{\sqrt{G_{n}(\omega)}}{Q(\omega)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}$ wherein ${\lambda_{o} = {\max\limits_{\omega \in B_{o}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}},$ and the computer processor is programmed to scale the transmit signal so that its energy is made equal to E_(o) above; and wherein the computer processor is programmed to determine the performance for the transmit signal F_(o)(ω) with transmit energy E_(o) and bandwidth B_(o), receiver signal H_(o)(ω) above given by ${SINR}_{o} = {\frac{1}{2\pi}{\int_{- B_{o}}^{B_{o}}{\frac{{{Q(\omega)}}^{2}{{F_{o}(\omega)}}^{2}}{{{G_{c}(\omega)}{{F_{o}(\omega)}}^{2}} + {G_{n}(\omega)}}\ {{\mathbb{d}\omega}.}}}}$ wherein Q(ω) is the target signal Fourier transform; G_(c)(ω) is the interference spectrum; and G_(n)(ω) is the noise spectrum.
 14. The apparatus of claim 12 wherein the computer processor is programmed to select a second bandwidth B_(C) smaller than the given bandwidth B_(o) so as to satisfy the condition ${{{SINR}_{1}\left( {E_{c},B_{c}} \right)} = {{a_{c} - \frac{c_{c}^{2}}{E_{c} + b_{c}}} = {SINR}_{o}}},$ where the second energy level E_(c) smaller than the energy E_(o) is given by $E_{c} = {\frac{1}{2\pi}{\int_{- B_{c}}^{B_{c}}{\sqrt{G_{n}(\omega)}\left( {\lambda_{c} - \frac{\sqrt{G_{n}(\omega)}}{Q(\omega)}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}$ with ${\lambda_{c} = {\max\limits_{\omega \in B_{c}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}}},{and}$ ${a_{c} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{c})}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},{b_{c} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{c})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},{and}$ ${c_{c} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{c})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}\ {\mathbb{d}\omega}}}}},$ where Ω₊(B_(c)) represents the frequency band (−B_(c)≦ω≦B_(c)).
 15. The apparatus of claim 14 further comprising wherein the computer processor is programmed to select a third bandwidth B_(D) smaller than the second bandwidth B_(C) and solving for a third energy E_(F) so as to satisfy the condition ${{{SINR}_{1}\left( {E_{F},B_{D}} \right)} = {{a_{D} - \frac{c_{D}^{2}}{E_{F} + b_{D}}} = {SINR}_{o}}},{and}$ ${a_{D} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{D})}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{\mathbb{d}\omega}}}}},{b_{D} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{D})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\mathbb{d}\omega}}}}},{and}$ ${c_{D} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{D})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\mathbb{d}\omega}}}}},$ where Ω₊(B_(D)) represents the frequency band (−B_(D)≦ω≦B_(D)).
 16. An apparatus comprising a transmitter; a receiver; and a computer processor; wherein the computer processor is programmed to cause the transmitter to output a first transmit signal having a first transmit signal waveform; wherein the computer processor is programmed to cause the transmitter to output a second transmit signal having a second transmit signal waveform; wherein the first transmit signal is different from the second transmit signal and the first transmit signal waveform is different from the second transmit signal waveform; wherein the first transmit signal has a prescribed bandwidth which is B_(o); wherein the first transmit signal has a prescribed energy which is E_(o); wherein the second transmit signal has a smaller bandwidth which is B_(C); wherein the second transmit signal has a smaller energy level which is E_(c); wherein the first transmit signal has a transform F_(o)(ω) which is determined by a user input into the computer processor; wherein the second transmit signal has transform F₁(ω) whose magnitude function is given by ${{F_{1}(\omega)}}^{2} = \left\{ {\begin{matrix} {{\sqrt{G_{n}(\omega)}\left( {\left( {\max\limits_{\omega \in B_{c}}\frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right) - \frac{\sqrt{G_{n}(\omega)}}{{Q(\omega)}}} \right)\frac{{Q(\omega)}}{G_{c}(\omega)}},} & {{- B_{c}} \leq \omega \leq B_{c}} \\ {0,} & {otherwise} \end{matrix};} \right.$ and wherein the first and second transmit signals have the same performance index SINR_(o) given by ${SINR}_{o} = {\frac{1}{2\pi}{\int_{- B_{o}}^{B_{o}}{\frac{{{Q(\omega)}}^{2}{{F_{o}(\omega)}}^{2}}{{{G_{c}(\omega)}{{F_{o}(\omega)}}^{2}} + {G_{n}(\omega)}}{\mathbb{d}\omega}}}}$ in terms of target detection in interference and noise, when used in conjunction with a receiver filter for the receiver such that the receiver filter has a Fourier transform H_(opt)(ω) given by ${H_{opt}(\omega)} = {\frac{{Q^{*}(\omega)}{F^{*}(\omega)}}{{{G_{c}(\omega)}{{F(\omega)}}^{2}} + {G_{n}(\omega)}}{\mathbb{e}}^{{- {j\omega}}\; t_{o}}}$ wherein t_(o) is a decision instant at which the target signal is to be detected, and either F₁(ω) or F_(o)(ω) are substituted for F(ω) above accordingly.
 17. The apparatus of claim 16 wherein the computer processor is programmed to cause the transmitter to generate a time domain signal that is constant envelope substantially everywhere while maintaining the transform magnitude |F₁(ω)| by starting with a signal f_(o)(t) that has energy E_(o) and bandwidth B_(o) and the computer processor is programmed to iteratively compute f _(n+1)(t)=P _(c) {P _(B) _(c) (f _(n)(t))}, n=0,1,2 . . . where $\left. {P_{B_{e}}\left( {f_{n},(t)} \right)}\leftrightarrow\left\{ {\begin{matrix} {{{{F_{1}(\omega)}}{\mathbb{e}}^{{- j}\;{\psi_{n}{(\omega)}}}},} & {{\omega } \leq B_{c}} \\ {0,} & {{\omega } > B_{c}} \end{matrix},} \right. \right.$ with ⇄ representing the standard Fourier transform operation, ψ_(n)(ω) the phase function of the Fourier transform of f_(n)(t), and ${P_{c}\left\{ {f(t)} \right\}} = \left\{ {\begin{matrix} {f(t)} & {{{f(t)}} \leq c} \\ {{+ c},} & {{f(t)} > c} \\ {{- c},} & {{f(t)} < {- c}} \end{matrix},} \right.$ and wherein the computer processor is programmed to determine the constant envelope level c above by normalizing the final iterated signal f_(n+1)(t) to maintain the prescribed energy level E_(c); and wherein the normalized iterate f_(n+1)(t) gives the second transmit signal waveform, and wherein the second transmit signal waveform maintains constant envelope substantially everywhere.
 18. The apparatus of claim 16 wherein the computer processor is programmed to cause the transmitter to output a third transmit signal having a third transmit signal waveform; wherein the third transmit signal is different from the first and second transmit signals and the third transmit signal waveform is different from the first and second transmit signal waveforms; wherein the third transmit signal has a prescribed bandwidth which is B_(D) that is smaller than the second transmit signal bandwidth B_(C); wherein the third transmit signal has an energy which is E_(F) that is greater than the second transmit signal energy E_(C), but smaller than the first transmit signal energy E_(o); wherein the third transmit signal has a Fourier transform F₂(ω) whose magnitude function is given by ${{F_{2}(\omega)}}^{2} = \left\{ {{\begin{matrix} {\frac{\sqrt{G_{n}(\omega)}\left( {{{\lambda\left( E_{F} \right)}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)},} & {\omega \in {Q_{+}\left( B_{D} \right)}} \\ {0,} & {otherwise} \end{matrix}{wherein}{\lambda\left( E_{F} \right)}} = {\frac{E_{F} + {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{D})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\mathbb{d}\omega}}}}}{\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{D})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\mathbb{d}\omega}}}}.}} \right.$ and wherein the third, second and first transmit signals have the same performance index SINR_(o) in terms of target detection in interference and noise, when the third signal is used in conjunction with a receiver filter that has a Fourier transform H_(opt)(ω) given by ${H_{opt}(\omega)} = {\frac{{Q^{*}(\omega)}{F_{2}^{*}(\omega)}}{{{G_{c}(\omega)}{{F_{2}(\omega)}}^{2}} + {G_{n}(\omega)}}{\mathbb{e}}^{{- {j\omega}}\; t_{o}}}$ wherein t_(o) is a decision instant at which the target signal is to be detected.
 19. The apparatus of claim 18 wherein the computer processor is programmed to cause the transmitter to generate a time domain signal that is substantially constant envelope while maintaining the third transform magnitude |F₂(ω)| by starting with a signal f_(o)(t) that has the required energy and bandwidth and the computer processor is programmed to iteratively compute f _(n+1)(t)=P _(c) {P _(B) _(D) (f _(n)((t))}, n=0,1,2 . . . wherein $\left. {P_{B_{D}}\left( {f_{n}(t)} \right)}\leftrightarrow\left\{ {\begin{matrix} {{{{F_{2}(\omega)}}{\mathbb{e}}^{- {{j\psi}_{n}{(\omega)}}}},} & {{\omega } \leq B_{D}} \\ {0,} & {{\omega } > B_{D}} \end{matrix},} \right. \right.$ with ⇄ representing the standard Fourier transform operation, ψ_(n)(ω) the phase function of the Fourier transform of f_(n)(t), and ${P_{c}\left\{ {f(t)} \right\}} = \left\{ {\begin{matrix} {{f(t)},} & {{{f(t)}} \leq c} \\ {{+ c},} & {{f(t)} > c} \\ {{- c},} & {{f(t)} < {- c}} \end{matrix},} \right.$ and wherein the computer processor determines the constant envelope level c above by normalizing the final iterated signal f_(n+1)(t) to maintain the prescribed energy level E_(F) wherein the normalized iterate signal f_(n+1)(t) gives the third transmit signal waveform and wherein the third transmit signal waveform maintains substantially constant envelope.
 20. The apparatus of claim 18 wherein the computer processor is programmed to select a fourth bandwidth B_(G) smaller than the third bandwidth B_(D) so as to satisfy the condition ${{{SINR}_{1}\left( {E_{o},B_{G}} \right)} = {{a_{G} - \frac{c_{G}^{2}}{E_{o} + b_{G}}} = {SINR}_{o}}},$ where E_(o) corresponds to the prescribed energy, and ${a_{G} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{G})}}{\frac{{{Q(\omega)}}^{2}}{G_{c}(\omega)}{\mathbb{d}\omega}}}}},{b_{G} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{G})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\mathbb{d}\omega}}}}},{and}$ ${c_{G} = {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{G})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\mathbb{d}\omega}}}}},$ where Ω₊(B_(G)) represents the frequency band (−B_(G)≦ω≦B_(G)).
 21. The apparatus of claim 20 wherein the computer processor is programmed to cause the transmitter to construct a fourth transmit signal having a fourth transmit signal waveform; wherein the fourth transmit signal is different from the first, second and third transmit signals and the fourth transmit signal waveform is different from the first, second and third transmit signal waveforms; wherein the fourth transmit signal has a bandwidth which is B_(G) that is smaller than the third transmit signal bandwidth B_(D); wherein the fourth transmit signal has energy E_(o) that is the same as the first transmit signal energy; wherein the fourth transmit signal has transform F₃(ω) whose magnitude function is given by ${{F_{3}(\omega)}}^{2} = \left\{ {{\begin{matrix} {\frac{\sqrt{G_{n}(\omega)}\left( {{{\lambda\left( E_{o} \right)}{{Q(\omega)}}} - \sqrt{G_{n}(\omega)}} \right)}{G_{c}(\omega)},} & {\omega \in {\Omega_{+}\left( B_{G} \right)}} \\ {0,} & {otherwise} \end{matrix}{wherein}{\lambda\left( E_{o} \right)}} = {\frac{E_{o} + {\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{G})}}{\frac{G_{n}(\omega)}{G_{c}(\omega)}{\mathbb{d}\omega}}}}}{\frac{1}{2\pi}{\int_{\Omega_{+}{(B_{G})}}{\frac{\sqrt{G_{n}(\omega)}{{Q(\omega)}}}{G_{c}(\omega)}{\mathbb{d}\omega}}}}.}} \right.$ and wherein the fourth, third, second and first transmit signals have the same performance index SINR_(o) in terms of target detection in interference and noise, when used in conjunction with the receiver filter for the receiver such that the receiver filter has a Fourier transform H_(opt)(ω) given by ${H_{opt}(\omega)} = {\frac{Q*(\omega)F_{3}*(\omega)}{{{G_{c}(\omega)}{{F_{3}(\omega)}}^{2}} + {G_{n}(\omega)}}{\mathbb{e}}^{{- {j\omega}}\; t_{o}}}$ wherein t_(o) is a decision instant at which the target signal is to be detected.
 22. The apparatus of claim 21 wherein the computer processor is programmed to generate a time domain signal that is substantially constant envelope while maintaining the given fourth transform magnitude |F₃(ω)| by starting with any suitable signal f_(o)(t) and iteratively computing f _(n+1)(t)=P _(c) {P _(B) _(G) (f _(n)(t))}, n=0,1,2 . . . wherein $\left. {P_{B_{G}}\left( {f_{n}(t)} \right)}\leftrightarrow\left\{ {\begin{matrix} {{{{F_{3}(\omega)}}{\mathbb{e}}^{- {{j\psi}_{n}{(\omega)}}}},} & {{\omega } \leq B_{G}} \\ {0,} & {{\omega } > B_{G}} \end{matrix},} \right. \right.$ with ⇄ representing the standard Fourier transform operation, ψ_(n)(ω) the phase function of the Fourier transform of f_(n)(t), and ${P_{c}\left\{ {f(t)} \right\}} = \left\{ {\begin{matrix} {{f(t)},} & {{{f(t)}} \leq c} \\ {{+ c},} & {{f(t)} > c} \\ {{- c},} & {{f(t)} < {- c}} \end{matrix},} \right.$ and determining the constant envelope level c above by normalizing the final iterated signal f_(n+1)(t) to maintain the prescribed energy level E_(o). The normalized iterate f_(n+1)(t) gives the desired fourth transmit signal waveform that maintains substantially constant envelope. 